# Strong h-Convexity and Separation Theorems.

1. Introduction

Let X be a normed space, let D be a convex subset of X, and let c > 0. A function f: D [right arrow] R is called strongly convex with modulus c (see, e.g., [1, 2]) if

f(tx + (1 - t)y) [less than or equal to] tf(x) + (1 - t) f(y) -ct(1 - t)[[parallel]x - y[parallel].sup.2] (1)

for all x, y [member of] D and t [member of] [0,1]. Recall also that the usual notion of convex functions corresponds to the case c = 0. Strongly convex functions, introduced by Polyak [3], play an important role in optimization theory and mathematical economics. Many properties and applications of them can be found in the literature (see, e.g., [2, 4-7] and the references therein).

In [8] Varosanec introduced the notion of h-convexity. Let h : [0,1] [right arrow] [R.sub.+] be a given function. A function f: D [right arrow] R is said to be h-convex if

f(tx + (1 - t)y) [less than or equal to] h(t) f(x) + h(1 - t) f(y) (2)

for all x, y [member of] D and t [member of] [0,1]. This notion unifies and generalizes the known classes of convex functions, s-convex functions, Godunova-Levin functions, and P-functions, which are obtained by putting in (2) h(t) = t, h(t) = [t.sup.s] (where s [member of] (0,1)), h(t) = 1/t (with h(0) = 0), and h(t) = 1, respectively. Some properties of them can be found, for example, in [8-13].

Combining the above two ideas we say that a function f: D [right arrow] R is strongly h-convex with modulus c (cf. [14]) if

f(tx + (1 - t) y) [less than or equal to] h(t) f(x) + h(1 - t) f(y) -ct(1 - t)[[parallel]x - y[parallel].sup.2] (3)

for all x, y [member of] D and t [member of] [0,1].

In this note we present a Jensen-type inequality for such functions and give a characterization of pairs of functions that can be separated by a strongly h-convex one. Separation (or sandwich) theorems, that is, theorems providing conditions under which two given functions can be separated by a function from some special class, play an important role in many fields of mathematics and have various applications. In the literature one can find numerous results of this type (see, e.g., [12,15-24]).

2. Jensen-Type Inequality

In the whole paper we assume that is a real inner product space (i.e., the norm [parallel] * [parallel] in X is induced by an inner product: [parallel]x[parallel] = [square root of (<x | x>))]. D is a convex nonempty subset of X and c is a positive constant.

A function h: [0,1] [right arrow] R is said to be multiplicative if

h(st) = h(s)h(t), s, t [member of] [0,1]. (4)

Note that if h is multiplicative, then it is nonnegative and either h = 0 or h(1) = 1. In what follows we assume that h [not equal to] 0.

The following result is a counterpart of the classical Jensen inequality for strongly h-convex functions. It generalizes the Jensen-type inequality for strongly convex functions obtained in [4]. Similar results for h-convex functions are proved in [8,12].

Theorem 1. Let h : [0,1] [right arrow] R be a multiplicative function such that h(t) [greater than or equal to] t for all t [member of] [0,1]. If a function f: D [right arrow] R is strongly h-convex with modulus c, then

[mathematical expression not reproducible], (5)

for all n [member of] N, [x.sub.1], ..., [x.sub.n] [member of] D, and [t.sub.1], ..., [t.sub.n] > 0 with [t.sub.1] + ... + [t.sub.n] = 1 and [bar.x] = [t.sub.1][x.sub.1] + ... + [t.sub.n][x.sub.n].

Proof. For n =1 inequality (5) is trivial and for n = 2 it follows from the definition of strong h-convexity (note that [t.sub.1][[parallel][x.sub.1] - [bar.x][parallel].sup.2] + [t.sub.2] [[parallel][x.sub.2] - [bar.x][parallel].sup.2] = [t.sub.1][t.sub.2][[parallel][x.sub.1] - [x.sub.2][parallel].sup.2]). Now, assuming (2) holds for some n, we will prove it for n + 1. By the definition of strong h-convexity we get

[mathematical expression not reproducible], (6)

where

[mathematical expression not reproducible]. (7)

By the inductive assumption we have

[mathematical expression not reproducible]. (8)

Now, using the above inequalities, the multiplicativity of h, and the assumption h(t) [greater than or equal to] t, we obtain

[mathematical expression not reproducible] (9)

To finish the proof it is enough to show that

[mathematical expression not reproducible], (10)

or, equivalently,

[mathematical expression not reproducible]. (11)

Since

[mathematical expression not reproducible], (12)

we have

[mathematical expression not reproducible], (13)

which finishes the proof.

3. Separation by Strongly h-Convex Functions

It is proved in [15] that two functions f, g :D [right arrow] R defined on a convex subset D of a vector space can be separated by a convex function if and only if

f([n.summation over (i=1)][t.sub.i][x.sub.i]) [less than or equal to] [n.summation over (i=1)] [t.sub.i]g([x.sub.i]), (14)

for all n [member of] N, [x.sub.1], ..., [x.sub.n] [member of] D, and [t.sub.1], ..., [t.sub.n] > 0 with [t.sub.1] + ... + [t.sub.n] =1.

In this section we present counterparts of that result related to strong h-convexity.

Theorem 2. Let f, g:D [right arrow] R be given functions and h : [0,1] [right arrow] R be a multiplicative function such that h(t) [greater than or equal to] t for all t [member of] [0,1]. If there exists a function f : D [right arrow] R strongly h-convex with modulus c such that

f [less than or equal to] [phi] [less than or equal to] g on D, (15)

then

[mathematical expression not reproducible], (16)

for all n [member of] N, [x.sub.1], ..., [x.sub.n] [member of] D, and [t.sub.1], ..., [t.sub.n] > 0 with [t.sub.1] + ... + [t.sub.n] = 1 and [bar.x] = [t.sub.1][x.sub.1] + ... + [t.sub.n][x.sub.n].

Proof. By the Jensen inequality for strongly h-convex functions (Theorem 1) we have

[mathematical expression not reproducible]. (17)

Theorem 3. Let f,g: D [right arrow] R be given functions and h : [0,1] [right arrow] R be a multiplicative function such that h(t) [less than or equal to] t for all t [member of] [0,1]. If

[mathematical expression not reproducible], (18)

for all n [member of] N, [x.sub.1], ..., [x.sub.n] [member of] D, and [t.sub.1], ..., [t.sub.n] > 0 with [t.sub.1] + ... + [t.sub.n] =1 and x = [t.sub.1][x.sub.1] + ... + [t.sub.n][x.sub.n], then there exists a function f : D [right arrow] R strongly h-convex with modulus c such that

f [less than or equal to] [phi] [less than or equal to] g on D. (19)

Proof. Fix x [member of] D and define a function f : D [right arrow] R by

[mathematical expression not reproducible]. (20)

By (18) the definition is correct and f(x) [less than or equal to] [phi](x) for all x [member of] D. On the other hand, taking n = 1 in the above definition (and, consequently, [t.sub.1] = 1, x = [x.sub.1]) and using the fact that h(1) = 1, we get [phi](x) [less than or equal to] g(x) for all x [member of] D.

To prove that [phi] is strongly h-convex with modulus c, fix x,y [member of] D and t [member of] [0,1]. Take arbitrary [u.sub.1], ..., [u.sub.n] [member of] D, [[alpha].sub.1], ..., [[alpha].sub.n] [member of] [0,1] and [v.sub.1], ..., [v.sub.m] [member of] D, [[beta].sub.1], ..., [[beta].sub.m] [member of] [0,1] such that [[alpha].sub.1] + ... + [[alpha].sub.n] = 1, [[beta].sub.1] + ... [[beta].sub.m] = 1 and x = [[alpha].sub.1] [u.sub.1] + ... + [[alpha].sub.n][u.sub.n], y = [[beta].sub.1][v.sub.1] + ... + [[beta].sub.m][v.sub.m]. Since [[summation].sup.n.sub.i=1] t[[alpha].sub.i] + [[summation].sup.m.sub.j=1] (1 - t) [[beta].sub.j] = 1, the point tx + (1 - t)y is a convex combination of [u.sub.1], ..., [u.sub.n], [v.sub.1], ..., [v.sub.m], and

tx + (1 - t)y = [n.summation over (i=1)] t[[alpha].sub.i] [u.sub.i] + [m.summation over (j=1)] (1 - t) [[beta].sub.j] [v.sub.j]. (21)

Therefore, by the definition of [phi] we have

[mathematical expression not reproducible], (22)

where m = tx +(1 - t)y. By the multiplicativity of h we have

h(t[[alpha].sub.i]) = h(t) h([[alpha].sub.i]),

h((1 - t)[[beta].sub.j]) = h(1 - t) h([[beta].sub.j]). (23)

Note also that

[mathematical expression not reproducible], (24)

and, similarly,

[mathematical expression not reproducible]. (25)

Hence, using the fact that h(t) [less than or equal to] t, we get

[mathematical expression not reproducible]. (26)

Substituting (23) and (26) into (22), we obtain

[mathematical expression not reproducible]. (27)

Now, taking the infimum in the first term and next in the second term of the right hand side of (27) and using the definition of [phi], we get

[phi] (tx +(1 - t) y) [less than or equal to] h (t) [phi](x) + h(1 - t)[phi](y) -ct(1 - t)[[parallel]x - y[parallel].sup.2], (28)

which shows that [phi] is strongly h-convex with modulus c and finishes the proof.

Remark 4. The method used in the proof of Theorem 3 is similar to that in [12, Theorem 3]. However, in our case we assume additionally that h(t) [less than or equal to] t, t [member of] [0,1]. The following example shows that this assumption is essential. Let h(t) = 1, t [member of] [0,1] and consider f, g : [0,1] [right arrow] R, f = -1 and g = 0. Then condition (18) is satisfied with c =1, but there is no function [phi] strongly h-convex (with any modulus) satisfying -1 = f [less than or equal to] [phi] [less than or equal to] g = 0. Indeed, if f is strongly h-convex with h = 1, then f is nonnegative and some of its values are positive (putting y = x in the definition of strong h-convexity, we get [phi] [greater than or equal to] 0, but [phi] = 0 is not strongly h-convex).

As a consequence of Theorem 3 we obtain the following Hyers-Ulam-type stability result for strongly h-convex functions.

Let [epsilon] be a positive constant. We say that a function g : D [right arrow] R is e-strongly h-convex with modulus c if

[mathematical expression not reproducible], (29)

for all n [member of] N, [x.sub.1], ..., [x.sub.n] [member of] D, and [t.sub.1], ..., [t.sub.n] > 0 with [t.sub.1] + ... + [t.sub.n] = 1 and [bar.x] = [t.sub.1][x.sub.1] + ... + [t.sub.n][x.sub.n].

Corollary 5. Let h : [0,1] [right arrow] R be a multiplicative function such that h(t) [less than or equal to] t for all t [member of] [0,1]. If a function g : D [right arrow] R is [epsilon]-strongly h-convex with modulus c, then there exists a function [phi] : D [right arrow] R strongly h-convex with modulus c such that

g(x) - [epsilon] < [phi](x) [less than or equal to] g(x), x [member of] D. (30)

Proof. Define f(x) = g(x) - [epsilon], x [member of] D. Then f and g satisfy (18). Therefore, by Theorem 3, there exists [phi] : D [right arrow] R, strongly h-convex with modulus c, such that g - [epsilon] [less than or equal to] [phi] [less than or equal to] g, on D.

http://dx.doi.org/10.1155/2016/7160348

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This research has been partially supported by Central Bank of Venezuela.

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Teodoro Lara, (1) Nelson Merentes, (2) and Kazimierz Nikodem (3)

(1) Departamento de Fasica y Matematicas, Universidad de los Andes, N. U. Rafael Rangel, Trujillo, Venezuela

(2) Universidad Central de Venezuela, Escuela de Matematicas, Caracas, Venezuela

(3) Department of Mathematics, University of Bielsko-Biala, Ul. Willowa 2, 43-309 Bielsko-Biala, Poland

Correspondence should be addressed to Kazimierz Nikodem; knikodem@ath.bielsko.pl

Received 9 July 2016; Accepted 16 October 2016